Preprint number: XXXX-XXXX Vortices and Other Topological Solitons in Dense Quark Matter Minoru Eto1, Yuji Hirono2;3;4, Muneto Nitta5y, and Shigehiro Yasui6 z 1Department of Physics, Yamagata University, Kojirakawa 1-4-12, Yamagata, Yamagata 990-8560, Japan 2Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan 3Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan 4Department of Physics, Sophia University, Tokyo 102-8554, Japan 5Department of Physics, and Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 6KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan ∗E-mail: [email protected] ............................................................................... Dense quantum chromodynamic matter accommodates various kind of topological solitons such as vortices, domain walls, monopoles, kinks, boojums and so on. In this review, we discuss various properties of topological solitons in dense quantum chro- modynamics (QCD) and their phenomenological implications. A particular emphasis is placed on the topological solitons in the color-flavor-locked (CFL) phase, which exhibits both superfluidity and superconductivity. The properties of topological solitons are dis- cussed in terms of effective field theories such as the Ginzburg-Landau theory, the chiral Lagrangian, or the Bogoliubov{de Gennes equation. The most fundamental string-like topological excitations in the CFL phase are the non-Abelian vortices, which are 1/3 quantized superfluid vortices and color magnetic flux tubes. These vortices are created at a phase transition by the Kibble-Zurek mechanism or when the CFL phase is realized in compact stars, which rotate rapidly. The interaction between vortices is found to be repulsive and consequently a vortex lattice is formed in rotating CFL matter. Bosonic and fermionic zero-energy modes are trapped in the core of a non-Abelian vortex and propagate along it as gapless excitations. The former consists of translational zero modes (a Kelvin mode) with a quadratic dispersion and CP 2 Nambu-Goldstone gapless modes with a linear dispersion, associated with the CFL symmetry spontaneously broken in the core of a vortex, while the latter is Majorana fermion zero modes belonging to the triplet of the symmetry remaining in the core of a vortex. The low-energy effective the- ory of the bosonic zero modes is constructed as a non-relativistic free complex scalar arXiv:1308.1535v2 [hep-ph] 2 Feb 2014 field and a relativistic CP 2 model in 1+1 dimensions. The effects of strange quark mass, electromagnetic interactions and non-perturbative quantum corrections are taken into account in the CP 2 effective theory. Various topological objects associated with non- Abelian vortices are studied; colorful boojums at the CFL interface, the quantum color magnetic monopole confined by vortices, which supports the notion of quark-hadron duality, and Yang-Mills instantons inside a non-Abelian vortex as lumps are discussed. The interactions between a non-Abelian vortex and quasi-particles such as phonons, glu- ons, mesons, and photons are studied. As a consequence of the interaction with photons, a vortex lattice behaves as a cosmic polarizer. As a remarkable consequence of Majo- rana fermion zero modes, non-Abelian vortices are shown to behave as a novel kind of non-Abelian anyon. In the order parameters of chiral symmetry breaking, we discuss fractional and integer axial domain walls, Abelian and non-Abelian axial vortices, axial wall-vortex composites, and Skyrmions. .............................................................................................. 1 typeset using TEX.cls PTP Subject Index xxxx, xxx y corresponding author z These authors contributed equally to this work 2 Contents PAGE 1 Introduction 6 2 Low-energy effective theories for high density QCD 14 2.1 Ginzburg-Landau theories 14 2.1.1 The CFL phase 14 2.1.2 Including strange quark mass 17 2.1.3 Including electromagnetic interactions 18 2.1.4 Time-dependent Ginzburg-Landau theory 20 2.2 Effective theories for light fields at zero temperature 21 2.2.1 Effective theory for U(1)B phonons at zero temperature 21 2.2.2 Chiral Lagrangian of the CFL mesons: pions and the η0 meson 22 3 Vortices 26 3.1 Abelian vortices 26 3.1.1 U(1)B superfluid vortices 26 3.1.2 U(1)A axial vortices 28 3.2 Non-topological color-magnetic fluxes 28 3.3 Non-Abelian vortices: stable color-magnetic flux tubes 29 3.3.1 Minimal (M1) non-Abelian vortices 29 3.3.2 Non-minimal (M2) non-Abelian vortices 34 3.3.3 Orientational zero modes of non-Abelian vortices 36 4 Dynamics of vortices 38 4.1 The translational zero modes (Kelvin modes) 38 4.1.1 The effective theory of translational zero modes 38 4.1.2 Magnus and inertial forces 40 4.2 Interaction between non-Abelian vortices 41 4.2.1 Intervortex force 41 4.2.2 Dynamics of two vortices and a vortex ring 43 4.3 Decays of U(1)B vortices and non-minimal M2 non-Abelian vortices 45 4.4 Colorful vortex lattices under rotation 47 4.4.1 Vortex formation and vortex lattices as a response to rotation in conventional superfluids 47 4.4.2 Colorful vortex lattices 49 4.4.3 Tkachenko modes 51 4.5 Superfluid vortices from relativistic strings 51 5 Dynamics of orientational zero modes 52 5.1 Low-energy effective theory of orientational zero modes 52 5.2 Effects of strange quark mass 55 5.3 Effects of electromagnetic fields 57 5.3.1 The effects on the electromagnetic coupling on non-Abelian vortices57 5.3.2 BDM vortex 59 5.3.3 CP 1 vortex 60 5.3.4 Pure color vortex 61 5.3.5 Magnetic fluxes 62 5.3.6 Quantum mechanical decay of the CP 1 vortices 63 3 5.3.7 Comparison with other potential terms 65 5.4 Quantum monopoles and the quark-hadron duality 66 5.5 Yang-Mills instantons trapped inside a non-Abelian vortex 72 6 Interactions of non-Abelian vortices with quasiparticles 74 6.1 Interaction with phonons and gluons 74 6.1.1 Dual action and vortex-quasiparticle interaction 74 6.1.2 Orientation dependence of the vortex-vortex interaction 76 6.2 Interaction with mesons 76 6.3 Interaction with electromagnetic fields 78 6.3.1 Coupling of orientation modes with electromagnetic fields 78 6.3.2 Scattering of photons off a vortex 79 6.3.3 Vortex lattice as cosmic polarizer 80 7 Colorful boojums at a CFL interface 83 7.1 What are boojums? 83 7.2 Colorful boojums at the CFL phase side 84 7.2.1 The shape of boojums 84 7.2.2 Formation of colorful monopoles with strange quark mass 84 7.3 Colorful boojums at the npe phase side 85 7.3.1 Matching condition 85 7.3.2 Magnetic fluxes 87 8 Fermions in vortices 89 8.1 The Bogoliubov-de Gennes equations and Majorana fermions for Abelian vortices 89 8.2 Bogoliubov-de Gennes formalism and Majorana fermions for non-Abelian vortices 92 8.3 Effective theory in 1 + 1 dimension along vortex string 95 8.4 The absence of supercurrent along a vortex 96 8.5 Index theorem 98 8.6 Topological superconductor 102 9 Non-Abelian exchange statistics of non-Abelian vortices 104 9.1 Exchange of vortices 104 9.2 Abelian vortices with Majorana fermions 105 9.3 Non-Abelian vortices with Majorana fermions 107 9.4 Abelian/Non-Abelian vortices with Dirac fermions 110 10Topological objects associated with chiral symmetry breaking 112 10.1Axial domain walls 112 10.1.1Fractional axial domain walls in the chiral limit 112 10.1.2Integer axial domain walls with massive quarks 113 10.2Linear sigma model 114 10.3Abelian and non-Abelian axial vortices 116 10.3.1Abelian axial vortices 116 10.3.2Non-Abelian axial vortices 116 10.4 Composites of axial domain walls and axial vortices 119 10.4.1Abelian and non-Abelian axial vortices attached by fractional axial domain walls in the chiral limit 119 4 10.4.2Abelian axial vortices attached by a composite axial domain wall with massive quarks 123 10.5Quantum decays of axial domain walls 124 10.5.1Decay of integer axial domain walls 124 10.5.2Decay of fractional axial domain walls 126 10.6Quantum anomalies and transport effects on topological defects 127 10.7Skyrmions as qualitons 129 11Topological objects in other phases 131 11.12SC phase 131 11.1.1U(1)A domain walls 131 11.1.2Color magnetic flux tubes 132 11.2CFL+K 133 11.2.1Superconducting strings and vortons 133 11.2.2Domain walls and drum vortons 135 12Summary and discussions 137 A Non-Abelian vortices in supersymmetric QCD 147 B Toric diagram 150 C Derivation of the low-energy effective theory of orientational zero modes 153 D Derivation of the dual Lagrangians for phonons and gluons 155 D.1 Low-energy effective theory of the CFL phase 155 D.2 The dual transformation 155 D.2.1 The dual transformation of massive gluons 155 D.2.2 The dual transformation of U(1)B phonons 157 D.3 The dual Lagrangian 158 E Derivation of fermion zero modes 159 5 1. Introduction Topological solitons are a subject of considerable interest in condensed matter physics [240]. Their properties have been studied extensively and it has been found that they play quite important roles phenomenologically. One such example is in superfluidity. Superfluidity emerges in a wide variety of physical systems such as helium superfluids [361] or ultra- cold atomic gases [273, 274, 354]. Superfluids are known to accommodate quantized vortices as topological solitons, which are important degrees of freedom to investigate the dynam- ics of superfluids [90, 352, 353, 361].